Intense quasiperiodic beam dynamics in accelerating system: mathematical model and optimization method

The paper is devoted to quasiperiodic intense beam dynamics is accelerating system. Particle interaction accounting is conducted with the use of well-known cloud-in-cell method. This numerical method is formalized, and mathematical model of space charge field intensity is suggested. Coulomb field intensity is presented in the form of integral over the domain of particle phase states. This permits beam evolution to be described by integro-differential equations. Controlled process quality is characterized by integral functional values. Beam dynamics optimization problem is formulated as trajectory ensemble control problem for dynamic system. Analytical expression of quality functional variation is obtained. It makes possible directed methods using for beam dynamics optimization.


Introduction
This paper is devoted to intense quasiperiodic beam dynamics modeling and optimization. Beam evolution mathematical model contains integral representation of beam space charge field.
In this paper the version of clouds-in-cells method is used for space charge field determination [7]. This convenient for computation and time-saving method is usually considered to be exclusively numerical. In the article the formalization of this method is performed and the analytical expression of Coulomb field intensity is obtained in the form of integral over particle phase states domain.
In the article presented we consider beam evolution to be a complex of bunch center motion and the motions of particles of bunch. Beam quasiperiodicity is taken into account. Controlled dynamic process is described by a system of integro-differential equations. Beam quality criterion is presented in the form of integral functional defined at beam trajectories. Beam dynamics optimization problem is formulated as trajectory ensemble control problem. An analytical representation of quality functional variation is obtained; it provides the possibility of directed optimization methods applying.

Beam dynamics equations
Consider a model of intense beam longitudinal dynamics in some accelerating system. We will investigate the processes at beam quasiperiod (i.e. one bunch evolution) and determine space charge field under the assumption of beam spatial periodicity. Particle beam is supposed to be axially symmetric. A bunch is presented by model particles moving in conducting tube of radius . Model particles are supposed to be "thick" disks (disks-clouds) of radius and thickness 2∆ (for resting particle).
Let us introduce independent variable = (where is the speed of light, is the time) and cylindrical coordinates , , ; may axis be aligned with channel axis. Let particle phase state be presented by vector ( , ), where is longitudinal coordinate, is reduced impulse.
Controlled beam evolution is considered to be a complex of bunch center motion and model particle motions. Bunch center phase state is characterized by vector ( , ). Hence longitudinal beam dynamics is described by the equations: with initial conditions (3) Here is model particle number, ( , ) is -th particle phase vector; is particle specific charge; ( ) , ( ) are the longitudinal intensity components characterizing respectively the effect of RF field and space charge field on the disk-cloud as a whole; is program control vector function.
Note that model particles are considered to be positively charged in space charge field accounting; ( ) is Coulomb field intensity created by positively charged particles.

Particle interaction account
Let some control function and value of independent variable be fixed. Let us describe -th model particle form-factor in laboratory reference frame by piecewise continuous even function ( − , ) taking nonzero values only at interval (− , ); 2 = 2∆ �1 + 2 � is cloud size.
Spatial quasiperiod length in laboratory reference frame is 2 = �1 + 2 ⁄ , where is RF field first harmonic wavelength.

Phase domain partitioning into elements
Let us enclose the set of model particles phase states in some rectangle and partition the rectangle into the elements , = 1, by any way. Suppose the elements to be connected compact subsets of positive measure. Assume to be the charge value contributed by -th model particle to -th element. Charge values of elements are = ∑ , = 1,
Let us approximate the action of the element (with all its periodic images) on -th model particle by the force presenting the action of uniformly charged "thick" disk with radius , thickness 2ℎ, charge ,center position at structure axis and reduced impulse (and all periodic images of this disk) on -th model particle. Note that 2ℎ = 2 ⁄ , is natural number, 10 ≤ ≤ .
We will describe the action of periodic beam on -th model particle by the intensity = ∑ =1 ⁄ , where is model particle charge assumed to be positive.

Space charge field intensity
The intensity expression is obtained on the basis of the formulae presented in [7]. The dependence 2 = 2 ( ) is taken into account. The result is: Here = 0 ⁄ is bunch charge value; 0 is average beam current; 0 is electric constant; = �1 + 2 , 0 ( ) and 1 ( ) are Bessel functions of the first kind of order 0 and 1 correspondingly, , = 1,2, … are the zeros of Bessel function 0 ( ). Notice that the formulae (4)-(6) are valid under the assumption | − | ≤ , = 1, ; in the case | − | > the coordinate should be replaced by its periodic image nearest to .

Particle phase density and integral Coulomb field representation
Now let us suppose particle distribution to be continuous; particle phase states compose the domain , . Consider a smooth function ( , , ) satisfying the condition: for any mode of rectangle partitioning. The relation (4)

Particular case
Let us partition the rectangle into identical cells by straight lines parallel to coordinate axes ( cells along axis and cells along axis, = ). Let , , = 1, , = 1, be the cell charges and � , �, = 1, , = 1, be the points chosen in the cells arbitrarily. Besides, let us neglect the difference between cell impulses and assume = , = 1, . In addition, suppose model particles to have the equal thickness = ∆ �1 + 2 ⁄ (in laboratory reference frame). In this case integral expression of space charge field intensity takes the form: Here 2ℎ = 2 ⁄ is the cell size along axis; = �1 + 2 ; the condition | − | ≤ is imposed in view of the assumption of beam periodicity (similarly to the formulae (4)-(6)).
Computational formula (4) is now represented as follows: �, = 1, . Now it is enough to introduce the grid on variable only, and , = 1, are the charges of grid cells. The formulae (12),(10)- (11) correspond to Coulomb field calculation with the use of clouds-in-cells method based on spatial quasiperiod partitioning into the cells [7,8].

Integro-differential beam dynamics model
Let us generalize beam dynamics model (1)-(3). Primarily, let us take into account integral representation (8) of space charge field.
Here ∈ [0, ] is independent variable; is a constant; , are phase vectors of systems (14) and (15) respectively; ( ) is a control; vector functions 1 , 1 , 2 are determined by the method of external fields modeling; vector function 2 is determined by the method of particle interaction account; ( , ) is phase density defined on system (14) trajectories; 0 is initial domain in phase space of the system (14); 0 ( ) is initial phase density; , = { = ( , 0 , ): 0 ∈ 0 }. All the functions in the equations (14)-(16) are supposed to be rather smooth to use mathematical optimization methods [1]. Control ( ) is supposed to be piecewise-continuous vector-function taking values in a compact. one can obtain the system (1)-(2), where ( ) is presented in terms of integral (9) and averaged value of external force is replaced by integral in accordance with (13). In addition, initial bunch center phase state is presented in integral form (see (3), (17)).
Following the approach [1,4], it is not hard to prove that smooth function ( , , ) satisfying the condition (7) for any mode of rectangle partitioning at any ∈ [0, ], satisfies the equation (16) with initial condition (18).